Thursday, Feb 2, 2017 - 3:30pm - Allen 14
CAM seminar
Efficient iterative one-shot methods for PDE-constrained optimation problems.
Dr. Jun Liu, Mathematics & Statistical Sciences, Jackson State University

Title:  Efficient iterative one-shot methods for PDE-constrained optimation problems.
Abstract:  PDE-Constrained optimization problems arise in many different scientific and engineering applications, such as computational fluid dynamics, inverse problems of PDE, and medical imaging. In this talk, I will present some efficient iterative methods for solving the discretized first-order optimality KKT system of PDE-Constrained optimization problems. I first briefly introduce finite difference discretization and multigrid method. Then, semismooth Newton (SSN) method is motivated to handle non-smooth and non-linear PDE constraints.

Putting all components together, we obtained several efficient iterative methods: 

1. For elliptic and parabolic PDE cases, we employed SSN method as the outer iterations and multigrid method (one V cycle) as inner solvers for the Jacobian systems. The obtained solver delivers competitive performance compared with the currently available solvers.

2. For hyperbolic wave PDE cases, the multigrid method fails to work. Instead, we proposed a new scheme in time and a robust preconditioner to accelerate the convergence of the Krylov subspace method (e.g., GMRES).

3. More recently, we proposed several domain decomposition algorithms in time to address the difficulty of huge system size from the KKT system of parabolic PDE-constrained optimization, which allows one-shot methods to be used on parallel computing platforms.

Numerical results will be shown to illustrate the effectiveness of our proposed algorithms.


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